Local coordinates
Encyclopedia
Local coordinates are measurement indices into a local coordinate system
or a local coordinate space
. A simple example is using house numbers to locate a house
on a street
; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.
Local systems exist for convenience. Continuing the geographic example, one could use latitude and longitude for all terrestrial locations, but unless one has a highly precise GPS device, this is impractical.
Local coordinate spaces are also useful for their ability to model independently transformable aspects of geometrical scenegraphs. When modeling a car, for example, it is desirable to describe the center of each wheel with respect to the car's coordinate system, but then specify the shape of each wheel in separate local spaces centered about these points. This way, the information describing each wheel can be simply duplicated four times, and independent transformations (e.g., steering rotation) can be similarly effected. The disadvantage is computational cost: the rendering system must access the higher-level coordinate system of the car and combine it with the space of each wheel in order to draw everything in its proper place.
Local coordinates also afford digital designers a means around the finite limits of numerical representation. The tread marks
on a tire, for example, can be described using millimetres by allowing the whole tire to occupy the entire range of numeric precision available. The larger aspects of the car, such as its frame, might be described in centimetres, and the terrain that the car travels on could be specified in meters.
In differential topology
, local coordinates on a manifold
are defined by means of an atlas of charts
. The basic idea behind coordinate charts is that each small patch of a manifold can be endowed with a set of local coordinates. These are collected together into an atlas, and stitched together in such a way that they are self-consistent on the manifold.
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
or a local coordinate space
Coordinate space
In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F. It can be defined as the product space of F over a finite index set.-Definition:...
. A simple example is using house numbers to locate a house
House
A house is a building or structure that has the ability to be occupied for dwelling by human beings or other creatures. The term house includes many kinds of different dwellings ranging from rudimentary huts of nomadic tribes to free standing individual structures...
on a street
Street
A street is a paved public thoroughfare in a built environment. It is a public parcel of land adjoining buildings in an urban context, on which people may freely assemble, interact, and move about. A street can be as simple as a level patch of dirt, but is more often paved with a hard, durable...
; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.
Local systems exist for convenience. Continuing the geographic example, one could use latitude and longitude for all terrestrial locations, but unless one has a highly precise GPS device, this is impractical.
Local coordinate spaces are also useful for their ability to model independently transformable aspects of geometrical scenegraphs. When modeling a car, for example, it is desirable to describe the center of each wheel with respect to the car's coordinate system, but then specify the shape of each wheel in separate local spaces centered about these points. This way, the information describing each wheel can be simply duplicated four times, and independent transformations (e.g., steering rotation) can be similarly effected. The disadvantage is computational cost: the rendering system must access the higher-level coordinate system of the car and combine it with the space of each wheel in order to draw everything in its proper place.
Local coordinates also afford digital designers a means around the finite limits of numerical representation. The tread marks
Tread Marks
Tread Marks is a 3D, third-person perspective, multiplayer-focused tank combat and racing computer game developed by Longbow Digital Arts. The game won the 2000 Independent Games Festival grand prize, later renamed to the Seumas McNally award in honor of the game's lead programmer who died on 21...
on a tire, for example, can be described using millimetres by allowing the whole tire to occupy the entire range of numeric precision available. The larger aspects of the car, such as its frame, might be described in centimetres, and the terrain that the car travels on could be specified in meters.
In differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, local coordinates on a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
are defined by means of an atlas of charts
Atlas (topology)
In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
. The basic idea behind coordinate charts is that each small patch of a manifold can be endowed with a set of local coordinates. These are collected together into an atlas, and stitched together in such a way that they are self-consistent on the manifold.